homotopy

The subject of algebraic topology is concerned with finding discrete, effectively computableinvariants which distinguish one topological space or continuousmap from another. A simple example is the degree of a map from the circle S1 to itself. If f: S1 → S1 is a continuousmap, there is a well-defined integer deg(f) which, intuitively speaking, counts the number of times the loop f winds around the target circle. (If f is smooth, deg(f) is easy to compute: it's the integral of the pulled-back angle form f∗dθ, divided by 2π.) But there are only countably many integers, and the space of continuousmaps from S1 to itself is very large (uncountable at least), so degree is certainly not a complete description of such a map. It turns out that the degree classifies maps from S1 to itself up to homotopy: two such maps have the same degree if and only if one of them is continuously deformable into the other. This situation reproduces itself throughout algebraic topology. There is essentially no hope of finding computableinvariants of spaces and maps if we want to detect the exact space or map with the invariant; there are just too many continuous maps to put in any kind of sensible algebraic structure. But if we agree to consider two maps the same if they are homotopic, that is, one of them is continuously deformable into the other, then there is a very rich theory which can distinguish many different spaces via algebraic invariants called homology groups and homotopy groups.

Formally, we say that continuousmaps f0, f1 from a space X to a space Y are homotopic if there is a continuousmap H: X × [0,1] → Y (called a homotopy from f0 to f1) which satisfies H(x, 0) = f0(x) and H(x, 1) = f1(x) for every x ∈ X. If X and Y are pointed spaces, that is we have fixed base points ∗X ∈ X and ∗Y ∈ Y, then we suppose also that H(∗X, t) = ∗Y for every t ∈ [0,1] (that is, H is a homotopy relative to ∗X). (The concept of a pointed space is a technical kludge which turns out to be almost mandatory in algebraic topology.) In general if A ⊂ X, a homotopy relative to A is one for which H(a, t) = a for every a ∈ A and t ∈ [0,1]. The relation of homotopy is conventionally denoted by a symbol which is not available in HTML; it looks like an equals sign where the top bar is a tilde, halfway between ∼ and ≅.